Formulas for Solutions

Suppose
that someone has already worked out the details,
so all that you have to do is read a formula
and make a solution. We can usually assume that
a solution is to be aqueous unless stated otherwise.
What about the concentration of the substance to
be added? Common ways of describing the concentrations
of solutions are weight-in-weight, weight-in-volume,
volume-in-volume, and molarity. Less commonly
used descriptions include normality and molality.
These formulas all have one thing in common. A
quantity of solute is measured out, mixed with
solvent, and the volume is brought to some final
quantity after the solute is completely dissolved.
That is, solutions are typically prepared volumetrically.
Because solutes add volume to a quantity of solvent,
this method of preparation of solutions is necessary
to ensure that an exact desired concentration is
obtained.

There are exceptions, of course. For example,
culture media for bacteria are typically made
up by adding a measured amount of powdered
medium to a measured volume of water. In such
cases it isn't critical that a precise concentration
be obtained, thus a weight-to-volume method
is appropriate, instead of weight-in-volume.

Weight/weight (w/w) solutions

Perhaps the easiest way to describe a solution
is in terms of weight-in-weight (w/w). The weight
of the solute relative to the weight of the
final solution is described as a percentage.
For example, suppose you have a dye that is soluble
in alcohol. Rather than write the instructions, “take
3 grams dye and mix with 97 grams absolute alcohol,” you
can describe the solutions simply as 3% dye in
absolute alcohol. The formula applies to any
volume of solution that might be required. Three
grams dye plus 97 grams alcohol will have final
weight of 100 grams, so the dye winds up being
3% of the final weight. Note that the final weight
is not necessarily equal to the final volume.

Aqueous weight-in-weight solutions are the easiest
to prepare. Since 1 milliliter of water weighs
one gram, we can measure a volume instead of
weighing the solvent. A very common use of w/w
formulas is with media for the culture of bacteria.
Such media come in granular or powdered form,
often contain agar, and often require heat in
order to dissolve the components. Microbiological
media, especially when they contain agar, are
difficult to transfer from one vessel to another
without leaving material behind. They coat the
surfaces of glassware, making quite a mess. Using
a w/w formula the media and water can be mixed,
heated, then sterilized, all in a single container.
For example, tryptic soy agar, a very rich medium
used for growing a variety of bacterial species,
comes with instructions to simply mix 40 grams
agar with one liter (equivalent to 1 kilogram)
of deionized water, without adjusting the final
volume. Very little material is wasted and there
is less of a mess.

Weight-in-volume (w/v) solutions

When we describe a concentration as a percentage
without specifying the type of formula, we imply
that the solution is to be made using the weight-in-volume
(w/v) method. As with w/w, weight-in-volume is
a simple type of formula for describing the
preparation of a solution of solid material
in a liquid solvent. This method can be used
to describe any solution, but is commonly used
for simple saline solutions and when the formula
weight of the solute is unknown, variable,
or irrelevant, which is often the case with
complex dyes, enzymes or other proteins. Solutions
that require materials from natural sources are
often prepared w/v because the molecular formula
of the substance is unknown and/or because
the substance cannot be described by a single
formula.

A one percent solution is defined as 1 gram of
solute per 100 milliliters final volume. For example,
1 gram of sodium chloride, brought to a final volume
of 100 ml with distilled water, is a 1% NaCl solution.
To help recall the definition of a 1% solution,
remember that one gram is the mass of one milliliter
of water. The mass of a solute that is needed in
order to make a 1% solution is 1% of the mass of
pure water of the desired final volume. Examples
of 100% solutions are 1000 grams in 1000 milliliters
or 1 gram in 1 milliliter.

Volume/volume (v/v) solutions

Volume-in-volume is another rather simple way
of describing a solution. We simply describe
the percent total volume contributed by the
liquid solute. As with the other types of formulas
used in biology, we assume that the solvent
is water unless some other solvent is specified.

V/v is often used to describe alcohol solutions
that are used for histology or for working with
proteins and nucleic acids. For example, 70% ethanol
is simply 70 parts pure ethanol mixed with water
to make 100 parts total. To make a liter of such
a solution we would start with 0.7 L absolute ethanol
and bring the final volume to 1 liter with water.
More often we might find ourselves with 95% alcohol.
To make a 70% solution from a 95% stock solution
requires a little more calculation. We will talk
about that in a bit, when we discuss how to make
dilutions.

Destaining of protein gels refers to the soaking
of a stained gel in acidified alcohol so as to
remove all dye that is not bound to proteins, revealing
the bands. A useful destaining solution consists
of 7% methanol, 10% acetic acid. This means using,
per liter of final solution, 100 ml pure (or “glacial”)
acetic acid and 70 ml methanol.

Molarity

A disadvantage of describing formulas as w/v
(%) is that the description says nothing about
the actual concentration of molecules in solution.
What if we want equal amounts of two chemicals
to be mixed together, so that for each molecule
of substance #1 there is a single molecule
of substance #2? The same amount in grams will
likely not contain the same number of molecules
of each substance. Another disadvantage of the
w/v method is that the same chemical can come in
many forms, so that the same amount in grams of
one form of the chemical contains a different amount
of it than another form. For example, you may work
with a chemical that can be in one of several forms
of hydration. Calcium chloride can be
purchased as a dry chemical in anhydrous form,
so that what you weigh out is nearly all pure calcium
chloride. On the other hand you may have a stock
of dry chemical that is hydrated with seven water
molecules per molecule of calcium chloride. The
same mass of this chemical will contain fewer molecules
of calcium chloride.

When we are interested in the actual
concentration of molecules of a chemical in solution,
it is better to have a universal measurement
that works regardless of how the chemical is supplied.
As long as the molecular weight (sometimes called
formula weight) is known, we can describe a solution
in the form of moles per liter, or simply molar
(M).

Working with formula weights

As with w/v solutions, we weigh out a specific
amount of chemical when making a molar solution.
Unlike w/v solutions, the amount to weigh depends
on the molecular weight (m.w.) of the substance
in grams per mole (g/mol). In order to calculate
the desired mass of solute you will need to
know the formula weight. Formula weights are
usually printed on the label and identified
by the abbreviation f.w. Formula weight is
the mass of material in grams that contains
one mole of substance, and may include inert
materials and/or the mass of water molecules
in the case of hydrated compounds. For pure
compounds the formula weight is the molecular
weight of the substance and may be identified
as such.

For example, the molecular weight of calcium chloride
is 111.0 grams per mole (g/mol), which is the same
as the formula weight if the material is anhydrous.
Calcium chloride dihydrate (CaCl2•2H2O) is
147.0 g/mol. For CaCl2•6H2O (hexahydrate)
the formula weight is 219.1 g/mol.

A hydrated compound is a compound that is surrounded
by water molecules that are held in place by hydrogen
bonds. The water molecules in a hydrated compound
become part of the solution when the material is
dissolved. Thus, 111.0 grams of anhydrous CaCl2,
147.0 grams of dihydrated CaCl2, or 219.1 grams
of CaCl2 hexahydrate in one liter final volume
all give a 1 mole per liter solution, abbreviated
1M.

Suppose that you need one liter of a solution
of 10 mM calcium chloride (10 millimolar, or 0.01
moles per liter), and suppose that you have only
CaCl2 dihydrate. To make your 10 mM solution you
weigh out 1/100 of the formula weight for dihydrated
CaCl2, which is 0.01 x 147.0 = 1.47 grams and bring
it to one liter.

Complications With Formula Weights

Perhaps you cannot find a formula weight on a
label or perhaps you are planning a protocol
and do not have the actual chemicals on hand.
You can calculate molecular weight from the
chemical formula with the aid of a periodic
table. You must keep in mind that when you
purchase the chemical the formula weight may
not be identical to the molecular weight. Suppose
that you have already determined how much to
weigh out based on the molecular weight, but
the formula weight is greater due to hydration
or the presence of inert material. Your remedy
is simply to multiply your calculated mass
by the ratio of formula weight to molecular
weight (or simply recalculate the weight needed).

For example, suppose that you need 10 grams of
pure CaCl2 (m.w. 111.0 g/mol), then discovered
that all you have is the hexahydrated form (CaCl2•6H2O,
f.w. 219.1 g/mol). Take 219.1 divided by 111.0
and multiply by 10. You need 19.7 grams of CaCl2•6H2O.

Materials are not always available in 100% pure
form. The description on the label might indicate
that the chemical is >99% pure. Such is often
the case with enzymes or other proteins that must
be purified from natural sources. Most of us do
not worry about purity if it is above 99%. Greater
precision might be important to analytical chemist,
for example, but is seldom needed in biological
applications. If there are significant impurities
or if you insist on being as precise as you can,
then calculate the amount of material you need
and divide by the fraction representing purity
of the substance. For example, if you need 10 grams
of pure substance A but what you have is 95% pure,
then divide 10 grams by 0.95 to get 10.5 gram (note
that the result has been rounded to a reasonable
level of precision).

Most chemicals tend to absorb water unless they
are kept desiccated, that is to some extent they
are hydroscopic. This problem should not be confused
with the state of hydration of a substances, which
refers to the direct association of water molecules
with molecules of the substance through hydorgen
bonding. Magnesium chloride is commonly used in
biological buffers, and is notoriously hydroscopic.
The formula weight does not include the added mass
of water that is absorbed from the atmosphere,
in fact the amount of contamination depends on
how long and under what conditions the chemical
has been shelved, especially with respect to humidity.
It is usually not practical to worry about water
content, since it is so difficult to control. If
precision is critical, then chemicals should be
maintained under desiccating conditions or used
immediately before they can absorb a significant
amount of water.

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Created by David R. Caprette (caprette@rice.edu), Rice University 20 May 05Updated 10 Aug 12